On the Number of Representations for Simple Lie Groups

TL;DR

This paper investigates the growth rate of the number of irreducible representations of simple complex Lie groups, establishing that the cumulative count up to dimension N grows linearly with N.

Contribution

It improves existing bounds and proves that the sum of the number of irreducible representations up to dimension N is on the order of N.

Findings

Established that R_N=O(N) for the sum of representations

Improved previous bounds on the growth rate

Provided new asymptotic estimate for representation counts

Abstract

The growth rate function $r_N$ counts the number of irreducible representations of simple complex Lie groups of dimension $N$. While no explicit formula is known for this function, previous works have found bounds for $R_N=\sum_{i=1}^Nr_i$. In this paper we improve on previous bounds and show that $R_N=O(N)$.